Functional equation for the zeta function
The functional equation for Riemann zeta function is a result due to analytic continuation of Riemann zeta function:
![\[\zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s)\]](http://latex.artofproblemsolving.com/7/8/3/78380ea7c8198c2fc2ec42ee52af5523d1e286e9.png)
There are multiple proofs for the functional equation for Riemann zeta function, and this page presents a lightweighted approach which merely relies on Fourier series that
![\[\frac12-\{x\}=\sum_{n=1}^\infty{\sin(2\pi nx)\over\pi n}\]](http://latex.artofproblemsolving.com/f/8/f/f8f45c27ac6b4f8f958b13d733947579e0c7a016.png)
